Universal scaling dependence of QCD energy loss from data driven studies
UUniversal scaling dependence of QCD energy loss from datadriven studies
P. Christiansen, K. Tywoniuk, and V. Vislavicius Division of Particle Physics, Lund University, Sweden Departament d’Estructura i Constituents de la Materia and Institut de Ci`encies del Cosmos,Universitat de Barcelona, Mart´ı i Franqu´es 1, ES-80 028 Barcelona, Spain (Dated: October 8, 2018)In this paper we study the energy loss of jets in the QGP via the nuclear mod-ification factor R AA for unidentified particles at high p T ( (cid:38)
10 GeV/ c ) in and outof the reaction plane of the collision. We argue that at such a high p T there areno genuine flow effects and, assuming that the energy loss is only sensitive to initialcharacteristics such as the density and geometry, find that R AA depends linearly onthe (RMS) length extracted from Glauber simulations. Furthermore we observe thatfor different centrality classes the density dependence of the energy loss enters as thesquare root of the charged particle multiplicity normalized to the initial overlap area.The energy loss extracted for RHIC and LHC data from the R AA is found to exhibita universal behavior. I. INTRODUCTION
One of the most stunning results fromthe heavy ion programs at RHIC and LHCis the quenching of jets and single-inclusivehadron spectra [1–3]. Being perturbativeprobes for which we can calculate the vac-uum baseline to high precision, jets are po-tentially excellent probes of the medium cre-ated in heavy ion collisions. Modifications,arising due to interactions with the hot anddense matter, are indeed expected to arise attimescales comparable to the lifetime of the medium are typically characterized in termsof elastic and radiative energy losses [4, 5],for recent reviews see, e.g., [6–8]. Presentlyour theoretical control of the jet fragmenta-tion is however limited. In particular, theimportance of modifications of the jet sub-structures due to the transverse medium res-olution was only recently pointed out [9]. Re-cent results from the LHC on the suppres-sion of single-inclusive hadrons and jets are inthis context challenging to reconcile with thecorresponding observations at RHIC [10] andcall for the refinement of present theoretical a r X i v : . [ h e p - ph ] F e b tools. Furthermore, at RHIC it is challeng-ing to reconcile both the data on the nuclearmodification factor, R AA , and azimuthal flow,characterized by v , at high p T within modelsbased on radiative QCD mechanisms [11, 12].We take this uncertainty at the theoreticallevel as an opportunity to make a data drivenstudy that we present here. Similar studieshave also been carried out previously in [13–15], see also [16] for more theoretically drivenstudies, and we will return to how they differfrom the present work in Section IV.One of the challenges of modeling theenergy loss is that the medium created inheavy ion collisions behaves as a perfect liq-uid. There are at least 2 major issues. Firstof all, both the geometry and the dynamicalexpansion of the medium introduce a compli-cation for the clean extraction of the trans-port properties of the medium. The longi-tudinal expansion of the medium causes theenergy density to decrease quickly with time(as the inverse of the proper time in theBjorken model [17]), and this could clearlyaffect the path length dependence of the en-ergy loss. Furthermore, the dynamics of themedium in the transverse plane signifies thatin non-central collisions there is an asymmet-ric expansion of the medium, where the ex-pansion in the reaction plane is larger thanout-of-plane. To first order the latter effect issupposed to be negligible, but various stud- ies have documented significant effects [18].Since we wish to pursue a data driven study,these effects cannot be handled without re-course to modeling and so we will focus oncharacterizing the energy loss in terms of ini-tial state observables. It is quite remarkablethat this seems to work very well.Secondly, the convincing signals of collec-tive behavior in A-A collisions hint at the ex-istence of a strongly coupled system. This, inturn, challenges the paradigm of using per-turbative methods to calculate the relevantdegrees of freedom for the jet-medium inter-actions. Our present study avoids these con-ceptual difficulties.One could worry that the measured R AA in and out of the reaction plane is signifi-cantly affected by flow. Let us try to ar-gue here that for p T > c this is inour opinion not very likely. Flow is typicallycharacterized by introducing a mass depen-dence. Both measurements of v [19] and the R AA [20] have shown that for p T > c there is little or no difference between re-sults for pions and protons. The triangularflow, characterized by the coefficient v , alsoseems to disappear in this p T region [19]. Asthe baryon to meson ratios are rather similarfrom RHIC energy ( √ s NN = 200 GeV) toLHC energies ( √ s NN = 2 .
76 TeV) [21] thisindicates that also for RHIC energies we needto have data for p T > c to eliminateflow effects. In our opinion, this allows ussafely to assume that both R AA and v at high p T are dominated by energy loss. The effectof residual flow would be an underestimate(overestimate) of the quenching contributionin (out) of plane. There are no indicationsfor such an effect in Fig. 2.At high energies the energy loss of a col-ored parton going through a colored mediumis expected to be dominantly radiative.Na¨ıvely one expects that radiative QCD en-ergy loss [22–26] increases quadratic withpath length, since this follows from the stim-ulated emission probability of a single hard gluon [27, 28]. These emissions are howeverrare and one should also account for multiple soft emissions. This changes the path lengthdependence of the characteristic p T shift ofthe medium-modified spectra so it becomeslinear [29], see Appendix B. A crucial pointof this paper is that the existing data allowsto disentangle more than simply the pathlength dependence of the suppression. As wewill show, this information has to be supple-mented by including a dependence on the en-ergy density. While our results will rely onsimple estimates of both of these quantities,for details see Section II, the agreement withthe data at two, widely separated energies ofRHIC and LHC represent a strong argumentfor the consistency of the interpretation ofenergy loss in ultrarelativistic heavy ion col- lisions. Finally we note that there is a sig-nificant p T dependence of the R AA . We shallignore this p T dependence in our quantitativestudies and focus on a common p T region of p T ≈
10 GeV/ c for LHC and RHIC. The scal-ing plots we show, in particular Fig. 3, doeshowever indicate that the scaling relations wefind are also valid at higher p T .The outline of the paper is as follows. InSection II we describe the data driven set-upand the method for extracting the energy loss(or p T shift) from the A-A spectra. We go onto present the obtained results and discussthem in Sections III and IV, respectively. Fi-nally, our conclusions are summarized in Sec-tion V. II. DATA DRIVEN SET-UP
Figure 1 illustrates the idea behind thestudies presented here. Based on Glaubersimulations of the participant distribution,two centrality classes are selected where wecan relate some properties in- vs. out-of-plane. In our case the selection was done onthe characteristic length, which we define asthe root-mean-squared (RMS) of the distri-bution. We can then compare the in- andout-of-plane R AA data for different centralityclasses where these properties will agree. InFig. 1 we have chosen two centrality classes,10-20 % and 30-40 %, where the in-plane X [fm] -10 -5 0 5 10 Y [f m ] -10-50510 -10 -5 0 5 10 Y [f m ] -10-50510 -10 -5 0 5 10 pa r t N / pa r t N d In-Out-(c) ] c [GeV/ T p AA R
10% - 20%
AA, in R
30% - 40%
AA, out R (d) FIG. 1. (Color online) The extracted partic-ipant distribution for two Glauber samples at √ s NN = 2 .
76 TeV (10%-20 % centrality (a) and30%-40 % centrality (b)) rotated so that the re-action plane coincides with the x -axis. The in-plane RMS of the former equals approximatelythe out-of-plane of the latter. Comparison ofthe participant distributions and the R AA for thetwo cases are shown in (c) and (d), respectively. RMS width of the former distribution, de-noted L in , is approximately equal to the out-of-plane RMS, denoted L out , of the latter, seethe top panels. In the lower-left panel ofFig. 1 we also demonstrate that the partic-ipant distribution is quite similar in the twocases.An important motivation behind such asimplified event selection is the fact that incentral collisions we expect the distribution of hard scatterings (binary collisions) to bemore narrowly distributed around the origin.In that way the path length of the two sam-ples should on the average be quite similar,but we note most importantly that the den-sity is quite different. Moreover, the trans-verse expansion could be much more signif-icant in-plane than out-of-plane and couldspoil the comparison. In our studies we findthat the latter effect can be neglected andthis is in fact also, as mentioned above, whatone would expect to first order from theoret-ical arguments.Once we have fixed the characteristiclength to be similar, it remains to include theeffect of the difference in energy density. Asit is clearly seen in the lower-right panel ofFig. 1, comparing the R AA for our examplecases for which the path lengths were equalin- and out-of-plane does not result in thesame amount of suppression. The overlap-ping participant distributions are reasonablydescribed by two-dimensional Gaussian dis-tributions, see lower-left panel of Fig. 1, andso we assign an area as A ≈ πL in L out . Thenwe assume that the characteristic energy den-sity ρ of the sample is given by ρ = K dN/dη πL in L out , (1)where K is a constant that is assumed to de-pend little on centrality and collision energy.In the following we always set K = 1 GeV/fmsuch to make ρ have the units GeV/fm . Asthis density is not normalized in a meaningfulway (because of the data driven nature of thisstudy) we will in the following use arbitraryunits (arb. units) in the plots. The pseudora-pidity distribution, dN/dη , have been takenfrom [30]. In Sec. IV where we introduce the-oretical estimates for comparison we will dis-cuss how one can normalize this properly toextract meaningful physics parameters. Thedefinition of ρ is inspired by Bjorken’s energydensity estimate and the observation that themean transverse energy per produced parti-cle does not change violently as a function ofcentrality or collisions energy [31].The LHC data on charged particle R AA and v used in this publication have beentaken from [32, 33]. CMS has published sim-ilar data [34, 35] but with coarser segmen-tation in centrality and p T , while ALICE v measurements does not cover centralitiesabove 50 % [19]. The R AA in- and out-of-plane used in our data driven analysis hasbeen obtained as R AA, in = R AA (1 + 2 v ) and R AA, out = R AA (1 − v ), respectively. The p T bins for the R AA and v results do not matchperfectly but the closest p T points have beenused and as both the R AA and v are onlyrather moderately varying at high p T we con-sider this a negligible effect. The error barsshown in the figures for R AA, in and R AA, out always include the full statistical and system- atic uncertainties added in quadrature fromboth the R AA and v . Normalization errorsfor R AA have been ignored as they are ex-pected to be directly correlated across cen-tralities (and to some degree also across beamenergies). When R AA, in and R AA, out is com-pared we assume in our interpretation thatthe relative systematic error is smaller thanshown. For the R AA one expects e.g. theefficiency and corrections to have similar sys-tematic errors and so there it seems a com-mon shift of R AA, in and R AA, out is expected.On the other hand for v a systematic shiftwould tend to shift R AA, in and R AA, out inopposite directions. A better understandingof this aspect can only be obtained by theexperiments.To extract information beyond merely thelevel of suppression of the spectra, we wouldlike to study the phenomenon of energy lossmore directly [12, 36]. To this aim we willassume that the spectra in p-p and A-A col-lisions can be described by a power-law witha similar exponent and that the differencecomes from the fact that the primordial p T ofthe partonic A-A spectrum has been shiftedto lower values due to energy loss in themedium. Note that the shift itself could be p T dependent. Explicitly, the p T shift is de-fined as ∆ p T ≡ p T,i − p T,m , where p T,i is themomentum of the parton prior to energy losswhile p T,m is the momentum of the hadron asmeasured in the detector. Then, following asimilar method as employed by PHENIX [36],the p T spectra of particles in a certain cen-trality class can be compared viad N pp d p T,i ( p T,i ) = (cid:12)(cid:12)(cid:12)(cid:12) d p T,m d p T,i (cid:12)(cid:12)(cid:12)(cid:12) R AA ( p T,m ) d N pp d p T,m ( p T,m ) , (2)where the first term is the Jacobian of thetransformation, see Appendix A for furtherdetails. Since we a priori cannot predictthe dependence of the shift, we explore twoextreme relations between p T,i and p T,m inEq. (2): p T independent absolute and rela-tive energy losses (see Appendix A for fur-ther details). In all figures the central valuefor the p T loss is the average of the two es-timates and the systematic uncertainty boxshows the actual difference. Here we stressthat the observed scaling patterns are not af-fected by the resulting variations in the pa-rameterization of ∆ p T .One can find several scaling variables fromthe orientation-dependent R AA alone since,e.g., the squared scaling variable will alsoalign the R AA . As an additional criterium wewill therefore demand that the extracted en-ergy loss is approximately linear in the scal-ing variable. III. RESULTS
Figure 2 shows a summary of the main re-sults from our studies of LHC data. In the [fm] L AA R (a) AA, in R AA, out
R c
11 GeV/ » T p [fm] L T , p / T p D (b) c
11 GeV/ » T p in-planeout-of-plane [arb. units] L r AA R (c) [arb. units] L r T , p / T p D (d) [arb. units] L r AA R (e) [arb. units] L r T , p / T p D (f) FIG. 2. (Color online) Example of scaling rela-tions for LHC data in arbitrary units. R AA vs. L (a), ρ / L (c), ρ / L (e) and extracted energyloss ∆ p T /p T vs. the same scaling variables (b,d, f) are shown for p T ≈
13 GeV/ c . We have in-cluded the uncertainty arising from the unknownfunctional form of ∆ p T as shaded boxes on thepoints in the right column, see Appendix A fordetails. left column we plot the R AA , while in theright one the p T shift divided by the primor-dial momentum, ∆ p T /p T,i . Both quantitiesare plotted vs. the respective scaling vari-able, for which we explore three possibilities:the path length, L , in the uppermost row,then ρ / L in the center and finally ρ / L in the lower column. The motivation be-hind these choices will be discussed furtherin Sec. IV. The plots in the left column illus-trate that it is possible to find several scalingvariables for the R AA , but that the energyloss is only approximately linear for the scal-ing variable in the middle panel. Extrapolat-ing down, it even seems to vanish for L = 0,as expected. We thus find that all R AA and v values for a given p T can be described interms of a linear energy loss relation. ] c [GeV/ T p AA R (a) 10% - 20% AA, in R
20% - 30%
AA, out R ] c [GeV/ T p AA R (b) 30% - 40% AA, in R
50% - 60%
AA, out R ] c [GeV/ T p AA R (c) 40% - 50% AA, in R
60% - 70%
AA, out R ] c [GeV/ T p AA R (d) 50% - 60% AA, in R
70% - 80%
AA, out R FIG. 3. (Color online) The comparison between R AA in- and out-of-plane for situations wherethe scaling variable ρ / L is approximately thesame. As can be seen, the good agreement ob-served in Fig. 2 is reproduced at higher p T . Furthermore, in Fig. 3 we demonstrate that the proposed scaling variable, ρ / L ,seems to work reasonably well for all p T .Whereas the agreement is good for centralcollisions, one observes some tension for the70–80 % centrality class. In the most periph-eral collisions it is known that the differencebetween the reaction plane and the impactparameter plane is the largest so that oneis more sensitive to the description of indi-vidual collisions in the model. The impactof hard scatterings on the experimental mea-surement of v could also be significant dueto the smaller number of participants.In the remainder of this section we willshow that the scaling variable found abovealso works surprisingly well both at RHICand LHC. Recently PHENIX has publishedthe R AA vs. the event plane at very high p T for π [12]. One should note that p T spectrain p-p at RHIC and LHC are power law-likefor p T > c , but that the power law ex-ponent is quite different in the two cases. Therelationship between R AA and energy loss atLHC and RHIC is therefore different even ifthe R AA are quite compatible for each of thecentralities. The main change in the scal-ing variable going from LHC down to RHICenergies is an almost centrality independentdecrease of particle density dN/dη of a fac-tor 0.48 [30]. In our picture one thereforeexpects the energy loss to be approximately40% larger at LHC than at RHIC for simi- [arb. units] L r AA R AA, in R PbPb
AA, out R PbPb
AA, in R AuAu
AA, out R AuAu (a) [arb. units] L r T , p / T p D (b) PbPbAuAu FIG. 4. (Color online) R AA in- and out-of-planefor p T ∼ −
13 GeV/ c at √ s NN = 2 .
76 TeV(red points) and for p T >
10 GeV/ c at √ s NN =200 GeV (blue points) as a function of ρ / L (a).The corresponding p T shifts as a function of thesame scaling variable are shown in (b). Due tothe different shape of the p-p spectrum the en-ergy loss is the same in our model even if the R AA is different. lar centralities. This is very similar to whatwas found in [12]. Figure 4 demonstrates thatwhile the R AA as a function of the proposedscaling variable, ρ / L , is different at LHCand RHIC, see the left panel, the derived en-ergy losses (which takes into account the dif-ference in the power law exponents) fall on asingle curve as a function of the scaling vari-able, see the right panel. We have fitted the p T shift using two parameterizations. Thedeviation from a linear relation is only mod-est.
IV. DISCUSSION
In the sections above, we have extracted aquite robust scaling law relating the charac-teristic p T shift of high p T hadronic spectrain A-A collisions to generic properties of thecollision, such as the multiplicity density andthe RMS of its distribution, that seems towork over an order of magnitude in collisionenergy. Despite the fact that these propertiesare quite inclusive and do not take account The two fits are a linear, ∆ p T /p T = Cξ , and a non-linear relation, found by solving d p T /p T = C d ξ ,where ξ = ρ / L and C is the slope parameter.The latter parameterization illustrates that the de-viation from the linear dependence on the scalingvariable ξ is consistent with a constant relative en-ergy loss. of the dynamical evolution of the system cre-ated in these collisions, the observed scalingsuggests a dominant and consistent mecha-nism underlying the physics of jet quenchingfrom RHIC to LHC.In the discussion of energy loss we havefocused on the very high p T data while inFig. 3 one clearly observes large differencesat lower p T ( (cid:46) c ). Some of thosecan be attributed to the typically much largerflow in-plane than out-of-plane. It is impor-tant to note that the good agreement at high p T shows that the density variation seems tobe pivotal for the quenching mechanism, seeFig. 2. This might suggest that the trans-verse expansion of the medium has little ef-fect on jet quenching, i.e., the dilution ofthe medium is canceled by the longer pathlength. This important issue certainly de-serves further studies.It is tempting to interpret the results fromSec. III in light of radiative energy loss, seeAppendix B for a brief review. Note firstlythat the na¨ıve identification of the p T shiftwith the mean energy loss taken by one-gluonemission, which would lead to ∆ p T ∼ ˆ qL ∼ ρ / L , cf. Eq. (B4), fails to produce a scal-ing, see the bottom-right panel of Fig. 2. Ac-counting for multi-gluon emissions and thebias due to the steeply falling parton spec-trum one rather expects ∆ p T ∼ ρ / L , cf.Eq. (B3), which is close to what we observe in the data. Similar studies have, as mentioned before,been carried out by Lacey et al. [13–15]. Themain difference from our work is that in theirstudies they do not take the density effect fordifferent centralities into account and theyobtain a single curve for R AA vs. path length L . But, as can be seen in the top-left panelof Fig. 2, this relation breaks down whenone studies R AA in- and out-of-plane. There-fore their results should be supplemented bythe additional information we have extractedhere. There are also important differences inthe physical pictures one extracts. Based ontheir findings they assert that jet quenchingfirst sets in after a time of ≈ c [13]. Inour analysis, the intercept in the right panelof Fig. 4 is consistent with zero suggestingthat the plasma formation time does not playa role for quenching.We point out that our improved datadriven analysis also allows to extract someinformation about the centrality dependenceof the quenching phenomenon. Presently,we will identify the extracted density ρ fromEq. (1) with the transport parameter for jetquenching averaged over the trajectory of thejets, (cid:104) ˆ q (cid:105) , in the context of radiative energy To study the expected p T behavior of the shift fromradiative processes, ∆ p T ∼ p / T goes beyond thescope of our present study. Centrality /f m ] > [ G e V q < c
11 GeV/ » T p PbPb
FIG. 5. The (cid:104) ˆ q (cid:105) as a function of centrality forLHC data extracted using Eq. 3. loss. Then, from Eq. B3, we find (cid:104) ˆ q (cid:105) = (cid:18) L ∆ p T p T (cid:19) np T π ¯ α , (3)where n is the power of the invariant p-pspectrum and ¯ α = α s C R /π ( C R being therelevant color factor), and we refer to Ap-pendix B for further details. Figure 5 dis-plays the resulting centrality behavior, with¯ α = 0 . p T = 11 GeV/ c . However,we note that this interpretation of the datadriven results introduces some conceptual is-sues. In fact, we expect both n and ¯ α to varywith the center-of-mass collision energy. Thereason for the variation of the latter quan-tity, comes about since at RHIC (LHC) weexpect the high p T particles to be fragmentsfrom dominantly quarks (gluons) implying a different color factor in ¯ α . The similarity be-tween RHIC and LHC in Fig. 4 therefore ap-pears accidental in this context. We recallthat the main motivation behind the datadriven study was to avoid these conceptualdifficulties. In our opinion, the most solidconclusion that can be drawn from Fig. 5is the decrease of (cid:104) ˆ q (cid:105) by roughly a factor 4from central to peripheral collisions dictatedby the √ ρ dependence.Albeit the data-driven analysis and sub-sequent interpretation both deal with staticquantities, and therefore are inherently con-sistent, a serious caveat of the interpretationin terms of radiative energy loss is the ne-glecting of the longitudinal expansion of themedium. This can be estimated by makinguse of the dynamical scaling law for ˆ q [39, 40].For a Bjorken-expanding medium the averagetransport parameter (cid:104) ˆ q (cid:105) is related to the ini-tial ˆ q measured at some initial proper time τ as (cid:104) ˆ q (cid:105) ∼ τ ˆ q /L . This, in turn, impliesthat the expected path length dependencedue to medium-induced radiative processeswould scale as ∼ L / , rendering it incom-patible with the extracted scaling behavior.Within our data driven approach, these ideasrather imply that the extracted values of theaverage transport parameter involves a sig-nificantly largerinitial ˆ q in the early stagesof the collision. A generic theory driven ap-proach to a wide array of energy loss scenar-1ios were presented in [37] in the context of aMonte-Carlo model which also includes real-istic nuclear geometry and couples to a hy-drodynamical model of the plasma, see also,e.g., [41] for similar efforts.The extraction of the p T loss is done forcharged particles while the quenching sup-posedly affects the spectra at the partonlevel. The charged particle p T spectrum athigh p T largely reflects leading particles andas we know from measurements at LHC thatleading particle fragments in quenched andunquenched jets share similar fractions of thejet p T [42], this approximation is probablynot so bad. Still it would be interesting tomake a similar study with jets. V. CONCLUSIONS
In this study the goal have been to dis-tance ourselves as far as possible from mod-els of jet quenching and rather by selectingsamples from different centrality classes withsimilar path lengths to be able to isolatethe density effect and then study the pathlength dependence. Surprisingly the methodworks very well and is in fact in reasonableagreement with theoretical considerations. Acritical question is how the longitudinal ex-pansion of the medium affects jet quenchingand this has tremendous impact on how onewould interpret the results in terms of e.g. the path length dependence.Finally we note that the exact same den-sity dependence observed for different cen-trality classes for LHC data is consistent withRHIC data indicating that the dense matterat RHIC and LHC has fundamentally similarproperties.
ACKNOWLEDGMENTS
PC wishes to express his gratitude tothe Swedish Research Council for financialsupport. KT is supported by a Juan dela Cierva fellowship and by the researchgrants FPA2010-20807, 2009SGR502, theConsolider CPAN project and FEDER.
Appendix A: How to estimate the p T shift The definition of the nuclear modificationfactor is R AA ( p T ) = d N AA (cid:14) d p T N coll d N pp (cid:14) d p T , (A1)where N coll represents the number of binarycollisions (the nuclear overlap function) forthe given centrality class estimated from theGlauber model (see, e.g., [43]). Following thestandard interpretation of the suppression ofhadron spectra in A-A collisions, we assumethat it arises due to a p T shift of the primor-2dial parton spectrum. We will therefore writed N AA ( p T )d p T = N coll d N pp ( p (cid:48) T = p T + δ p T )d p (cid:48) T (cid:12)(cid:12)(cid:12)(cid:12) d p (cid:48) T d p T (cid:12)(cid:12)(cid:12)(cid:12) , (A2)where we have made explicit for which p T value the spectrum is evaluated at and in-cluded the Jacobian of the transformation,which also can be written as d p (cid:48) T (cid:14) d p T =1+d δ p T (cid:14) d p T . Thus, the Jacobian differs fromunity if δ p T is a function of p T . Explicitly, thespectrum on the LHS of Eq. (A2) is measuredat a given p T , while p (cid:48) T on the RHS representsthe primordial momentum of the parton priorto energy loss. Thus, the master equation toextract the energy loss via the p T shift readsd N pp ( p (cid:48) T )d p (cid:48) T = R AA ( p T ) d N pp ( p T )d p T (cid:12)(cid:12)(cid:12)(cid:12) d p T d p (cid:48) T (cid:12)(cid:12)(cid:12)(cid:12) (A3)Having no a priori knowledge about the spe-cific form of δ p T that enters the Jacobian,we will parameterize it using two “extreme”cases:1. Firstly, we assume that p T = k p (cid:48) T ,where 0 < k < p (cid:48) T d N pp ( p (cid:48) T )d p (cid:48) T = R AA ( p T ) p T d N pp ( p T )d p T . (A4)2. Secondly, we assume a constant p T shift, δ p T = const. The Jacobian is sim-ply unity, and we get thatd N pp ( p (cid:48) T )d p (cid:48) T = R AA ( p T ) d N pp ( p T )d p T . (A5) Relevant cases, for which typically δ p T ∼ p α T where 0 < α < p T shifts estimated fromthese two cases will be averaged and the dif-ference will be indicated as a systematic un-certainty of the procedure. Appendix B: Radiative energy loss
For highly energetic probes the hot anddense medium is parameterized by one char-acteristic transport coefficient, the so-called ˆ q parameter which encodes the transverse mo-mentum broadening per unit length. Heuris-tically, this parameter scales with the en-ergy density ρ as ˆ q ∝ ρ / . The largestenergy that can be carried by a medium-induced gluon accumulates momentum alongthe whole path length of the medium and isusually defined as ω c ≡ ˆ qL /
2. The spec-trum of induced gluons per unit length reads[22, 27] ω d I d ω d L = ¯ α (cid:114) ˆ qω , (B1)for energies ω < ω c , where ¯ α ≡ α s C R /π .It follows that the energy loss caused by thesingle-gluon emission, given by − d E (cid:14) d L =¯ α ˆ qL , is dominated by the hard sector, ω ∼ To be precise, the spectrum in Eq. (B1) is regular-ized at a minimal energy marking the onset of theBethe-Heitler regime. ω c . One should on the other hand keep inmind that the number of gluons, given by N ( ω ) ∼ (cid:112) ¯ α ω c /ω , becomes large for softgluons, in particluar, when ω < ¯ α ω c .The quenching factor, which encodes thepartonic spectrum modified in the mediumprior to fragmentation, is defined as Q ( p T ) ≡ (cid:90) ∞ d (cid:15) D ( (cid:15) ) d σ vac ( p T + (cid:15) ) (cid:14) d p T2 d σ vac ( p T ) (cid:14) d p T2 , (B2)where D ( (cid:15) ) is the probability distribution ofenergy loss. Assuming independent gluonemissions it is simply given by a Poisson dis-tribution [29], but this premise can be im-proved upon by including, e.g., phase-spacelimitations [40] or energy-momentum conser-vation, see [44, 45]. These corrected distribu-tions give rise to more complex scaling trendsthan discussed below, but will be neglected inthe following. Presently we assume that theinvariant p-p is well described by a power lawspectrum with constant exponent n . Then,in the large- n approximation we recast thequenching factor as Q ( p T ) = exp( − nδ p T (cid:14) p T ),where δ p T is directly related to the p T shiftof the medium-modified parton spectrum asd σ med ( p T ) (cid:14) d p T2 = d σ vac ( p T + δ p T ) (cid:14) d p T2 . See [9] for a discussion of the validity of such an as-sumption. For our present purposes, the quenchingfactor serves as a good indicator of the parametricbehavior of the nuclear modification factor R AA . 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